The usual von Neumann entropy only takes the information of the eigenvalues into account: $S(\sum_j p_j |j\rangle \langle j |) = -\sum_j p_j \log p_j$.
It is invariant under unitary transformations of the state. However, one might imagine many situations in which the eigenbasis itself already carries information, e.g. if it is also the eigenbasis of the Hamiltonian.
One naive way of assigning information to the eigenbasis is as follows: I define a computational basis. The minimal number of bits required to fully specify the coefficients of an eigenbasis of the state with respect to the computational basis might be a measure for the information related to the eigenbasis. However, that information might usually be infinite.
Does there exist a standard quantum information entropy or measure that also takes the eigenbasis into account?